Question

Recall the equation for a circle with center (h,k)

and radius r. At what point in the first quadrant does the line with equation y=1.5x+4
intersect the circle with radius 4 and center (0, 4)?
Find x and y

Answer 1

the point of intersection is (2.6667, 7).

Explanation:

The equation for a circle with center (h,k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

So the equation for the circle with radius 4 and center (0, 4) is:

x^2 + (y - 4)^2 = 16

The line with equation y = 1.5x + 4 intersects the circle when the x and y values satisfy both equations. Substituting y = 1.5x + 4 into the equation for the circle, we get:

x^2 + (1.5x + 4 - 4)^2 = 16

Simplifying and solving for x, we get:

x^2 + (1.5x)^2 = 16

2.25x^2 = 16

x^2 = 16/2.25

x = ±2.6667

Since we are looking for the point in the first quadrant, we take the positive value of x. Substituting x = 2.6667 into the equation for the line, we get:

y = 1.5(2.6667) + 4

y = 7

Therefore, the point of intersection is (2.6667, 7).